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<div class="section">
<div class="titlepage"><div><div><h5 class="title">
<a name="math_toolkit.stat_tut.weg.nccs_eg.nccs_power_eg"></a><a class="link" href="nccs_power_eg.html" title="Tables of the power function of the chi2 test.">Tables
          of the power function of the chi<sup>2</sup> test.</a>
</h5></div></div></div>
<p>
            This example computes a table of the power of the χ<sup>2</sup>
test at the 5% significance
            level, for various degrees of freedom and non-centrality parameters.
            The table is deliberately the same as Table 6 from "The Non-Central
            χ<sup>2</sup> and F-Distributions and their applications.", P. B. Patnaik, Biometrika,
            Vol. 36, No. 1/2 (June 1949), 202-232.
          </p>
<p>
            First we need some includes to access the non-central chi squared distribution
            (and some basic std output of course).
          </p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">non_central_chi_squared</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">chi_squared</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">non_central_chi_squared</span><span class="special">;</span>

<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span>

<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
<span class="special">{</span>
</pre>
<p>
            Create a table of the power of the χ<sup>2</sup> test at 5% significance level, start
            with a table header:
          </p>
<pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"[table\n[[[nu]]"</span><span class="special">;</span>
<span class="keyword">for</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">lam</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span> <span class="identifier">lam</span> <span class="special">&lt;=</span> <span class="number">20</span><span class="special">;</span> <span class="identifier">lam</span> <span class="special">+=</span> <span class="number">2</span><span class="special">)</span>
<span class="special">{</span>
   <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"[[lambda]="</span> <span class="special">&lt;&lt;</span> <span class="identifier">lam</span> <span class="special">&lt;&lt;</span> <span class="string">"]"</span><span class="special">;</span>
<span class="special">}</span>
<span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"]\n"</span><span class="special">;</span>
</pre>
<p>
            (Note: the enclosing [] brackets are to format as a table in Boost.Quickbook).
          </p>
<p>
            Enumerate the rows and columns and print the power of the test for each
            table cell:
          </p>
<pre class="programlisting"><span class="keyword">for</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span> <span class="identifier">n</span> <span class="special">&lt;=</span> <span class="number">20</span><span class="special">;</span> <span class="special">++</span><span class="identifier">n</span><span class="special">)</span>
<span class="special">{</span>
   <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"[["</span> <span class="special">&lt;&lt;</span> <span class="identifier">n</span> <span class="special">&lt;&lt;</span> <span class="string">"]"</span><span class="special">;</span>
   <span class="keyword">for</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">lam</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span> <span class="identifier">lam</span> <span class="special">&lt;=</span> <span class="number">20</span><span class="special">;</span> <span class="identifier">lam</span> <span class="special">+=</span> <span class="number">2</span><span class="special">)</span>
   <span class="special">{</span>
</pre>
<p>
            Calculate the χ<sup>2</sup> statistic for a 5% significance:
          </p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">cs</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">chi_squared</span><span class="special">(</span><span class="identifier">n</span><span class="special">),</span> <span class="number">0.05</span><span class="special">));</span>
</pre>
<p>
            The power of the test is given by the complement of the CDF of the non-central
            χ<sup>2</sup> distribution:
          </p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">beta</span> <span class="special">=</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">non_central_chi_squared</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span> <span class="identifier">lam</span><span class="special">),</span> <span class="identifier">cs</span><span class="special">));</span>
</pre>
<p>
            Then output the cell value:
          </p>
<pre class="programlisting">         <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"["</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">beta</span> <span class="special">&lt;&lt;</span> <span class="string">"]"</span><span class="special">;</span>
      <span class="special">}</span>
      <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"]"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
   <span class="special">}</span>
   <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"]"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
<span class="special">}</span>
</pre>
<p>
            The output from this program is a table in Boost.Quickbook format as
            shown below.
          </p>
<p>
            We can interpret this as follows - for example if ν=10 and λ=10 then the
            power of the test is 0.542 - so we have only a 54% chance of correctly
            detecting that our null hypothesis is false, and a 46% chance of incurring
            a type II error (failing to reject the null hypothesis when it is in
            fact false):
          </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                    <p>
                      ν
                    </p>
                  </th>
<th>
                    <p>
                      λ=2
                    </p>
                  </th>
<th>
                    <p>
                      λ=4
                    </p>
                  </th>
<th>
                    <p>
                      λ=6
                    </p>
                  </th>
<th>
                    <p>
                      λ=8
                    </p>
                  </th>
<th>
                    <p>
                      λ=10
                    </p>
                  </th>
<th>
                    <p>
                      λ=12
                    </p>
                  </th>
<th>
                    <p>
                      λ=14
                    </p>
                  </th>
<th>
                    <p>
                      λ=16
                    </p>
                  </th>
<th>
                    <p>
                      λ=18
                    </p>
                  </th>
<th>
                    <p>
                      λ=20
                    </p>
                  </th>
</tr></thead>
<tbody>
<tr>
<td>
                    <p>
                      2
                    </p>
                  </td>
<td>
                    <p>
                      0.226
                    </p>
                  </td>
<td>
                    <p>
                      0.415
                    </p>
                  </td>
<td>
                    <p>
                      0.584
                    </p>
                  </td>
<td>
                    <p>
                      0.718
                    </p>
                  </td>
<td>
                    <p>
                      0.815
                    </p>
                  </td>
<td>
                    <p>
                      0.883
                    </p>
                  </td>
<td>
                    <p>
                      0.928
                    </p>
                  </td>
<td>
                    <p>
                      0.957
                    </p>
                  </td>
<td>
                    <p>
                      0.974
                    </p>
                  </td>
<td>
                    <p>
                      0.985
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      3
                    </p>
                  </td>
<td>
                    <p>
                      0.192
                    </p>
                  </td>
<td>
                    <p>
                      0.359
                    </p>
                  </td>
<td>
                    <p>
                      0.518
                    </p>
                  </td>
<td>
                    <p>
                      0.654
                    </p>
                  </td>
<td>
                    <p>
                      0.761
                    </p>
                  </td>
<td>
                    <p>
                      0.84
                    </p>
                  </td>
<td>
                    <p>
                      0.896
                    </p>
                  </td>
<td>
                    <p>
                      0.934
                    </p>
                  </td>
<td>
                    <p>
                      0.959
                    </p>
                  </td>
<td>
                    <p>
                      0.975
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      4
                    </p>
                  </td>
<td>
                    <p>
                      0.171
                    </p>
                  </td>
<td>
                    <p>
                      0.32
                    </p>
                  </td>
<td>
                    <p>
                      0.47
                    </p>
                  </td>
<td>
                    <p>
                      0.605
                    </p>
                  </td>
<td>
                    <p>
                      0.716
                    </p>
                  </td>
<td>
                    <p>
                      0.802
                    </p>
                  </td>
<td>
                    <p>
                      0.866
                    </p>
                  </td>
<td>
                    <p>
                      0.912
                    </p>
                  </td>
<td>
                    <p>
                      0.943
                    </p>
                  </td>
<td>
                    <p>
                      0.964
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      5
                    </p>
                  </td>
<td>
                    <p>
                      0.157
                    </p>
                  </td>
<td>
                    <p>
                      0.292
                    </p>
                  </td>
<td>
                    <p>
                      0.433
                    </p>
                  </td>
<td>
                    <p>
                      0.564
                    </p>
                  </td>
<td>
                    <p>
                      0.677
                    </p>
                  </td>
<td>
                    <p>
                      0.769
                    </p>
                  </td>
<td>
                    <p>
                      0.839
                    </p>
                  </td>
<td>
                    <p>
                      0.89
                    </p>
                  </td>
<td>
                    <p>
                      0.927
                    </p>
                  </td>
<td>
                    <p>
                      0.952
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      6
                    </p>
                  </td>
<td>
                    <p>
                      0.146
                    </p>
                  </td>
<td>
                    <p>
                      0.27
                    </p>
                  </td>
<td>
                    <p>
                      0.403
                    </p>
                  </td>
<td>
                    <p>
                      0.531
                    </p>
                  </td>
<td>
                    <p>
                      0.644
                    </p>
                  </td>
<td>
                    <p>
                      0.738
                    </p>
                  </td>
<td>
                    <p>
                      0.813
                    </p>
                  </td>
<td>
                    <p>
                      0.869
                    </p>
                  </td>
<td>
                    <p>
                      0.911
                    </p>
                  </td>
<td>
                    <p>
                      0.94
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      7
                    </p>
                  </td>
<td>
                    <p>
                      0.138
                    </p>
                  </td>
<td>
                    <p>
                      0.252
                    </p>
                  </td>
<td>
                    <p>
                      0.378
                    </p>
                  </td>
<td>
                    <p>
                      0.502
                    </p>
                  </td>
<td>
                    <p>
                      0.614
                    </p>
                  </td>
<td>
                    <p>
                      0.71
                    </p>
                  </td>
<td>
                    <p>
                      0.788
                    </p>
                  </td>
<td>
                    <p>
                      0.849
                    </p>
                  </td>
<td>
                    <p>
                      0.895
                    </p>
                  </td>
<td>
                    <p>
                      0.928
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      8
                    </p>
                  </td>
<td>
                    <p>
                      0.131
                    </p>
                  </td>
<td>
                    <p>
                      0.238
                    </p>
                  </td>
<td>
                    <p>
                      0.357
                    </p>
                  </td>
<td>
                    <p>
                      0.477
                    </p>
                  </td>
<td>
                    <p>
                      0.588
                    </p>
                  </td>
<td>
                    <p>
                      0.685
                    </p>
                  </td>
<td>
                    <p>
                      0.765
                    </p>
                  </td>
<td>
                    <p>
                      0.829
                    </p>
                  </td>
<td>
                    <p>
                      0.879
                    </p>
                  </td>
<td>
                    <p>
                      0.915
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      9
                    </p>
                  </td>
<td>
                    <p>
                      0.125
                    </p>
                  </td>
<td>
                    <p>
                      0.225
                    </p>
                  </td>
<td>
                    <p>
                      0.339
                    </p>
                  </td>
<td>
                    <p>
                      0.454
                    </p>
                  </td>
<td>
                    <p>
                      0.564
                    </p>
                  </td>
<td>
                    <p>
                      0.661
                    </p>
                  </td>
<td>
                    <p>
                      0.744
                    </p>
                  </td>
<td>
                    <p>
                      0.811
                    </p>
                  </td>
<td>
                    <p>
                      0.863
                    </p>
                  </td>
<td>
                    <p>
                      0.903
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      10
                    </p>
                  </td>
<td>
                    <p>
                      0.121
                    </p>
                  </td>
<td>
                    <p>
                      0.215
                    </p>
                  </td>
<td>
                    <p>
                      0.323
                    </p>
                  </td>
<td>
                    <p>
                      0.435
                    </p>
                  </td>
<td>
                    <p>
                      0.542
                    </p>
                  </td>
<td>
                    <p>
                      0.64
                    </p>
                  </td>
<td>
                    <p>
                      0.723
                    </p>
                  </td>
<td>
                    <p>
                      0.793
                    </p>
                  </td>
<td>
                    <p>
                      0.848
                    </p>
                  </td>
<td>
                    <p>
                      0.891
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      11
                    </p>
                  </td>
<td>
                    <p>
                      0.117
                    </p>
                  </td>
<td>
                    <p>
                      0.206
                    </p>
                  </td>
<td>
                    <p>
                      0.309
                    </p>
                  </td>
<td>
                    <p>
                      0.417
                    </p>
                  </td>
<td>
                    <p>
                      0.523
                    </p>
                  </td>
<td>
                    <p>
                      0.62
                    </p>
                  </td>
<td>
                    <p>
                      0.704
                    </p>
                  </td>
<td>
                    <p>
                      0.775
                    </p>
                  </td>
<td>
                    <p>
                      0.833
                    </p>
                  </td>
<td>
                    <p>
                      0.878
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      12
                    </p>
                  </td>
<td>
                    <p>
                      0.113
                    </p>
                  </td>
<td>
                    <p>
                      0.198
                    </p>
                  </td>
<td>
                    <p>
                      0.297
                    </p>
                  </td>
<td>
                    <p>
                      0.402
                    </p>
                  </td>
<td>
                    <p>
                      0.505
                    </p>
                  </td>
<td>
                    <p>
                      0.601
                    </p>
                  </td>
<td>
                    <p>
                      0.686
                    </p>
                  </td>
<td>
                    <p>
                      0.759
                    </p>
                  </td>
<td>
                    <p>
                      0.818
                    </p>
                  </td>
<td>
                    <p>
                      0.866
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      13
                    </p>
                  </td>
<td>
                    <p>
                      0.11
                    </p>
                  </td>
<td>
                    <p>
                      0.191
                    </p>
                  </td>
<td>
                    <p>
                      0.286
                    </p>
                  </td>
<td>
                    <p>
                      0.387
                    </p>
                  </td>
<td>
                    <p>
                      0.488
                    </p>
                  </td>
<td>
                    <p>
                      0.584
                    </p>
                  </td>
<td>
                    <p>
                      0.669
                    </p>
                  </td>
<td>
                    <p>
                      0.743
                    </p>
                  </td>
<td>
                    <p>
                      0.804
                    </p>
                  </td>
<td>
                    <p>
                      0.854
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      14
                    </p>
                  </td>
<td>
                    <p>
                      0.108
                    </p>
                  </td>
<td>
                    <p>
                      0.185
                    </p>
                  </td>
<td>
                    <p>
                      0.276
                    </p>
                  </td>
<td>
                    <p>
                      0.374
                    </p>
                  </td>
<td>
                    <p>
                      0.473
                    </p>
                  </td>
<td>
                    <p>
                      0.567
                    </p>
                  </td>
<td>
                    <p>
                      0.653
                    </p>
                  </td>
<td>
                    <p>
                      0.728
                    </p>
                  </td>
<td>
                    <p>
                      0.791
                    </p>
                  </td>
<td>
                    <p>
                      0.842
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      15
                    </p>
                  </td>
<td>
                    <p>
                      0.105
                    </p>
                  </td>
<td>
                    <p>
                      0.179
                    </p>
                  </td>
<td>
                    <p>
                      0.267
                    </p>
                  </td>
<td>
                    <p>
                      0.362
                    </p>
                  </td>
<td>
                    <p>
                      0.459
                    </p>
                  </td>
<td>
                    <p>
                      0.552
                    </p>
                  </td>
<td>
                    <p>
                      0.638
                    </p>
                  </td>
<td>
                    <p>
                      0.713
                    </p>
                  </td>
<td>
                    <p>
                      0.777
                    </p>
                  </td>
<td>
                    <p>
                      0.83
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      16
                    </p>
                  </td>
<td>
                    <p>
                      0.103
                    </p>
                  </td>
<td>
                    <p>
                      0.174
                    </p>
                  </td>
<td>
                    <p>
                      0.259
                    </p>
                  </td>
<td>
                    <p>
                      0.351
                    </p>
                  </td>
<td>
                    <p>
                      0.446
                    </p>
                  </td>
<td>
                    <p>
                      0.538
                    </p>
                  </td>
<td>
                    <p>
                      0.623
                    </p>
                  </td>
<td>
                    <p>
                      0.699
                    </p>
                  </td>
<td>
                    <p>
                      0.764
                    </p>
                  </td>
<td>
                    <p>
                      0.819
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      17
                    </p>
                  </td>
<td>
                    <p>
                      0.101
                    </p>
                  </td>
<td>
                    <p>
                      0.169
                    </p>
                  </td>
<td>
                    <p>
                      0.251
                    </p>
                  </td>
<td>
                    <p>
                      0.341
                    </p>
                  </td>
<td>
                    <p>
                      0.434
                    </p>
                  </td>
<td>
                    <p>
                      0.525
                    </p>
                  </td>
<td>
                    <p>
                      0.609
                    </p>
                  </td>
<td>
                    <p>
                      0.686
                    </p>
                  </td>
<td>
                    <p>
                      0.752
                    </p>
                  </td>
<td>
                    <p>
                      0.807
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      18
                    </p>
                  </td>
<td>
                    <p>
                      0.0992
                    </p>
                  </td>
<td>
                    <p>
                      0.165
                    </p>
                  </td>
<td>
                    <p>
                      0.244
                    </p>
                  </td>
<td>
                    <p>
                      0.332
                    </p>
                  </td>
<td>
                    <p>
                      0.423
                    </p>
                  </td>
<td>
                    <p>
                      0.512
                    </p>
                  </td>
<td>
                    <p>
                      0.596
                    </p>
                  </td>
<td>
                    <p>
                      0.673
                    </p>
                  </td>
<td>
                    <p>
                      0.74
                    </p>
                  </td>
<td>
                    <p>
                      0.796
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      19
                    </p>
                  </td>
<td>
                    <p>
                      0.0976
                    </p>
                  </td>
<td>
                    <p>
                      0.161
                    </p>
                  </td>
<td>
                    <p>
                      0.238
                    </p>
                  </td>
<td>
                    <p>
                      0.323
                    </p>
                  </td>
<td>
                    <p>
                      0.412
                    </p>
                  </td>
<td>
                    <p>
                      0.5
                    </p>
                  </td>
<td>
                    <p>
                      0.584
                    </p>
                  </td>
<td>
                    <p>
                      0.66
                    </p>
                  </td>
<td>
                    <p>
                      0.728
                    </p>
                  </td>
<td>
                    <p>
                      0.786
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      20
                    </p>
                  </td>
<td>
                    <p>
                      0.0961
                    </p>
                  </td>
<td>
                    <p>
                      0.158
                    </p>
                  </td>
<td>
                    <p>
                      0.232
                    </p>
                  </td>
<td>
                    <p>
                      0.315
                    </p>
                  </td>
<td>
                    <p>
                      0.402
                    </p>
                  </td>
<td>
                    <p>
                      0.489
                    </p>
                  </td>
<td>
                    <p>
                      0.572
                    </p>
                  </td>
<td>
                    <p>
                      0.648
                    </p>
                  </td>
<td>
                    <p>
                      0.716
                    </p>
                  </td>
<td>
                    <p>
                      0.775
                    </p>
                  </td>
</tr>
</tbody>
</table></div>
<p>
            See <a href="../../../../../../example/nc_chi_sq_example.cpp" target="_top">nc_chi_sq_example.cpp</a>
            for the full C++ source code.
          </p>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
</div>
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